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Theorem divval 10234
Description: Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divval
Distinct variable groups:   ,   ,

Proof of Theorem divval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4155 . . 3
2 eqeq2 2472 . . . . 5
32riotabidv 6259 . . . 4
4 oveq1 6303 . . . . . 6
54eqeq1d 2459 . . . . 5
65riotabidv 6259 . . . 4
7 df-div 10232 . . . 4
8 riotaex 6261 . . . 4
93, 6, 7, 8ovmpt2 6438 . . 3
101, 9sylan2br 476 . 2
11103impb 1192 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  \cdif 3472  {csn 4029  iota_crio 6256  (class class class)co 6296   cc 9511  0cc0 9513   cmul 9518   cdiv 10231
This theorem is referenced by:  divmul  10235  divcl  10238  cnflddiv  18448  divcn  21372  rexdiv  27622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-div 10232
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